Methods for indexing solid forms of compounds

ABSTRACT

The methods of the invention determine the unit cell parameters of a crystalline solid form using diffraction data and applying an algorithm. Using the algorithm, the unit cell parameters may be determined, which may allow one to distinguish between different crystalline solid forms of a substance.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. §119 to U.S.Provisional Application 61/238,941, filed Sep. 1, 2009, which isincorporated herein by reference.

TECHNICAL FIELD

The invention described herein relates to methods of characterizingsolid forms, and methods of determining the unit cell of a crystallinesolid form which may provide information about the solid form, forexample to distinguish between different crystalline solid forms of asubstance.

BACKGROUND

Diffraction is an experimental technique in which radiation with anarrow range of wavelengths is shone on a sample. The radiationinteracts with the electrons and/or nuclei in the sample and isscattered elastically. Interference within the scattered radiationcreates an observable pattern that is characteristic of themolecular-scale structure of the sample. To be effective, the radiationshould have a wavelength that is similar to the atomic scale. Forexample, X-rays, high energy electrons, and thermal neutrons may beused. X-rays are readily produced using laboratory sources and arenon-destructive at sufficiently low doses. Therefore, X-ray diffraction(XRD) is the most commonly used diffraction technique.

Single crystal diffraction is used to determine the molecular scalestructure of a crystalline form of matter using a single crystal.Diffracted radiation is measured as a function of scattering angle andrelative orientation of the crystal. Typically thousands of coherentpeaks in the interference pattern are used to determine the size andshape of the crystal unit cell and the positions of each atom in thecrystal. While very useful, the need for a large crystal with fewdefects limits the application of single crystal diffraction.

If only a powder sample (as opposed to a single crystal) is available,then powder diffraction may be used to obtain a sub-set of theinformation available using single crystal diffraction. For a samplecomposed of many tiny and randomly oriented particles, the resultingdiffraction pattern has continuous rings instead of discrete points andthe diffraction pattern is largely independent of the orientation of thesample. Due to overlap of the rings, typically dozens of coherent peaksin the interference pattern are measured as a function of scatteringangle. If X-rays are used for the radiation and the sample is a powder,then the technique is called X-ray powder diffraction (XRPD). FIG. 1provides a schematic view of XRPD. As noted above, alternate forms ofradiation may also be used.

Although the invention set forth herein will be described primarily withrespect to XRPD, it should be understood that other forms of powderdiffraction, such as electron and neutron powder diffraction, can alsobe used.

The term “crystalline” as used herein includes polycrystalline,microcrystalline, nanocrystalline, and partially or wholly crystallinesubstances, as well as disordered crystalline substances. Crystallinesolid forms can include, for example, cocrystals, solvates and hydrates.Crystalline solid forms can also include polymorphs, which are differentcrystalline solid forms having the same chemical composition.Crystalline solid forms can include crystalline forms of salts ofcompounds, for instance, salts of pharmaceutical compounds. In addition,as used herein reference to a crystalline solid form of “a compound”includes a crystalline solid form comprising a compound and optionallyone or more additional compounds or components, i.e., a multi-componentsystem. For instance, a crystalline solid form of a compound includes acocrystal and includes a salt of a compound.

The result of XRPD is an XRPD pattern. For samples composed ofcrystalline powders, the XRPD pattern may be a combination of sharppeaks and broad features. Sharp peaks are due to ordered crystallineregions in the sample. The sharp peaks occur at particular scatteringangles (2θ) relative to the transmitted X-ray beam. The broad featuresmay be due to a variety of factors, such as disorder in the sample,defects in the crystal, and/or scattering by air.

The XRPD pattern is a measured intensity (I) as a function of scatteringangle (2θ). The positions of the peaks in the pattern and the relativeintensities of the peaks are characteristic of a well-prepared sample ofa particular crystalline material. Therefore, an XRPD pattern identifiesa particular material, just as a fingerprint identifies a particularperson. For many purposes, the XRPD pattern can be used directly withoutadditional analysis.

Powder diffraction data may also be used to determine thecrystallographic unit cell of the crystalline structure. Many methods ofpowder diffraction indexing are known. XRPD indexing, and in factindexing of all powder diffraction data, is the process of determiningthe size and shape of the crystallographic unit cell consistent with thepeak positions in a given XRPD pattern. Indexing does not make use ofthe relative intensity information in the XRPD pattern. The goal of theindexing process is the determination of three unit cell lengths(a,b,c), three unit cell angles (α,β,γ), and three Miller index labels(h,k,l) for each peak. The lengths are typically reported in Angstromunits (Å), and the angles in degree units. The Miller index labels areunitless integers. Successful indexing indicates that the sample iscomposed of one crystalline phase and is therefore not a mixture ofcrystalline phases. The indexing solution also provides a concise meansto convey the positions of allowed peaks in an XRPD pattern. Otherexemplary methods of powder diffraction indexing include, by way ofexample, Dicvol and X-cell, both of which are known to those of skill inthe art.

Crystallographic unit cells are not unique. For any crystallinematerial, there is an infinite set of unit cells that may be used todescribe the crystal structure of the material. There is, however, aunique unit cell called the reduced basis that has a minimal volume (V)and whose parameters (a,b,c,α,β,γ) conform to a set of rules. Thereduced basis provides a systematic means for categorizing and comparingcrystal forms. Reduced bases conform to one of 44 tabulated“characters.” Characters are a means of classification of reduced unitcells. Lattices belong to the same character if their reduced cells canbe continuously deformed into one-another without changing the Bravaistype and with continuous changes of the reduced lattice parameters. Foreach character there is a tabulated matrix transformation that generatesa conventional unit cell from the reduced basis. The conventional cellparameters are most often reported as the outcome of XRPD indexing.

The invention makes use of an algorithm developed by the inventor,herein termed the “Triads Algorithm.” By applying the Triads Algorithmto measured instrumental data, the benefits of the invention, asdescribed below, may be achieved.

In the Triads Algorithm described below, the overall strategy is toidentify a member of the infinite set of unit cells that describe thelattice, then use well-established methods to transform the cell to itsreduced basis and finally to the conventional unit cell.

The Bragg Equation (1) relates the Bragg angle (θ) to the order of thereflection (n), the radiation wavelength (λ), and the distance betweenMiller planes (d_(hkl)):nλ=2d _(hkl) sin(θ)  (1)Rearranging equation (1) provides an equation for the scattering vectormagnitude (κ_(hkl)):

$\begin{matrix}{\kappa_{hkl} = {\frac{2\;\sin\;(\theta)}{\lambda} = \frac{n}{d_{hkl}}}} & (2)\end{matrix}$Although the symbol k is often used for the magnitude of the scatteringvector, here the symbol κ is used to avoid confusion with the Millerindices (h,k,l). The distance between Miller planes (d_(hkl)) is afunction of the Miller indices (h,k,l) and the crystallographic unitcell parameters (a,b,c,α,β,γ). This function is most convenientlyexpressed using matrix multiplication:

$\begin{matrix}{\left( \kappa_{hkl} \right)^{2} = {\left\lbrack {h\mspace{14mu} k\mspace{14mu} l} \right\rbrack \cdot \underset{\underset{\_}{\_}}{B} \cdot \begin{bmatrix}h \\k \\l\end{bmatrix}}} & (3)\end{matrix}$where B is the Bragg matrix:

$\begin{matrix}{\underset{\underset{\_}{\_}}{B} = \begin{bmatrix}\left( a^{*} \right)^{2} & {\left( a^{*} \right)\left( b^{*} \right){\cos\left( \gamma^{*} \right)}} & {\left( a^{*} \right)\left( c^{*} \right){\cos\left( \beta^{*} \right)}} \\{\left( a^{*} \right)\left( b^{*} \right){\cos\left( \gamma^{*} \right)}} & \left( b^{*} \right)^{2} & {\left( b^{*} \right)\left( c^{*} \right){\cos\left( \alpha^{*} \right)}} \\{\left( a^{*} \right)\left( c^{*} \right){\cos\left( \beta^{*} \right)}} & {\left( b^{*} \right)\left( c^{*} \right){\cos\left( \alpha^{*} \right)}} & \left( c^{*} \right)^{2}\end{bmatrix}} & (4)\end{matrix}$and (a*,b*,c*,α*,β*,γ*) are reciprocal cell parameters. Equations (3)and (4) constitute the quadratic form of the Bragg Equation. It givesthe peak position for a given Miller index triplet (h,k,l) given a unitcell of specified reciprocal cell parameters.

Simplified versions of the quadratic form are available for symmetriccrystal systems. These simplifications are special cases of the generaltriclinic case provided in equations (3) and (4). Since the tricliniccase is general, it is used in the implementation of the TriadsAlgorithm described below. Higher symmetry unit cells are recognizedduring the course of the algorithm.

Friedel's Law states that peaks labeled with Miller indices (h,k,l) and(−h,−k,−l) are indistinguishable in the absence of absorption effects.Slight deviations from Friedel's Law are used in the context of singlecrystal diffraction to determine absolute configuration of chiralmolecules, but this is not relevant to XRPD since the opposing pairs ofpeaks overlap each other in an XRPD pattern. Since Friedel pairs alwaysoverlap in XRPD patterns, it is convenient to choose a naming conventionthat eliminates this ambiguity. In a preferred embodiment of the presentinvention, all Miller indices may be chosen such that the first non-zeroindex is positive. This embodiment is used in the examples below. In afurther embodiment, all Miller indices may be chosen such that the firstnon-zero index is negative. In yet a further embodiment, a combinationof positive and negative first non-zero Miller indices may be chosen.

Each of the infinite set of unit cells that describe a particularcrystalline material yields reflections at the same set of peakpositions (2θ) through the Bragg Equation. Larger unit cells willindicate additional peaks that are not indicated for a minimal volumeunit cell, however. Since different unit cells have different unit cellparameters, the Miller indices labeling each peak are also different,but the set of distances between Miller planes is fixed. An exampleillustrating the equivalence of different unit cells is given in TABLE1.

TABLE 1 Exemplary comparison of Miller index labels for three differentunit cells describing the same crystal with the same peak positions.Arbitrarily Specified Cell a = 10.000 Å, b = 17.321 Å, Reduced BasisConventional Cell c = 26.458 Å, α = 10.89° a = b = c = 10.000 Å, a = b =c = 14.142 Å, Observed Peak β = 55.46° γ = 54.74° α = β = γ = 60° α = β= γ = 90° Positions (2θ) (Triclinic) (Rhombohedral) (Face-CenteredCubic) 10.827° (100), (123), (011), (012) (001), (010), (100), (111)(111) 12.508° (112), (023), (111) (011), (101), (110) (002), (020),(200) 17.724° (1-1-1), (223), (135), (01-1), (10-1), (1-10), (022),(202), (220) (134), (1-1-2), (001) (112), (121), (211) 20.815° (212),(034), (1-2-3), (11-1), (1-11), (1-1-1), (113), (131), (311) (2, 3, 5),(101), (146), (021), (201), (210), (035), (124), (234), (012), (102),(120), (211), (122), (10-1) (122), (212), (221) 21.752° (200), (246),(022), (024) (002), (020), (200), (222) (222) 25.168° (224), (046),(222) (022), (202), (200) (004), (040), (400)For each of three different unit cells, Miller indices are given foreach of the observed peaks. In TABLE 1 there are multiple Miller indicescorresponding to each peak. This is a result of multiple lattice planesthat diffract at the same Bragg angle and therefore overlap in the XRPDpattern. Such coincidence of reflections is common for symmetric unitcells such as the one used in TABLE 1. Each column in TABLE 1 is adifferent description of the same unit cell, but with different unitcell parameters and Miller index labels for the observed peaks. Thisdemonstrates that there are multiple indexing solutions (column 1, forinstance), any one of which may be reduced to the reduced basis (column2) and then transformed to the conventional cell (column 3). This allowsone to assign Miller indices in a convenient fashion, for example duringthe generation of trial solutions during the Triads Algorithm. Once theAlgorithm generates a unit cell that is consistent with the observedpeak list, then the cell can be reduced and transformed to theconventional cell.

SUMMARY

The invention described herein relates to the characterization ofcrystalline solid forms. In accordance with exemplary embodiments of theinvention, the inventor has discovered novel methods for determining theunit cell parameters of a crystalline solid form in a process known asindexing. In various embodiments of the invention, one or morediffraction methods are used to obtain data for a crystalline solidform, an algorithm is applied to obtain unit cell parameters, and thepeaks in the XRPD pattern are indexed.

One exemplary embodiment of the invention is a method for determiningthe crystal unit cell parameters of a crystalline solid form, comprisinggenerating an X-ray powder diffraction pattern of a solid crystallinesubstance and determining the unit cell parameters of the substance byperforming the Triads Algorithm to identify one or more sets of valuesof unit cell parameters of the crystalline solid form. Further exemplaryembodiments may include one or more refinement steps.

In at least one embodiment, the methods of the invention may be applied,for example, to distinguish between different crystalline solid forms ofa substance.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying figures illustrate various exemplary embodiments of theinvention and, together with the description, serve to furtherillustrate certain principles of the invention.

FIG. 1 illustrates a schematic view of X-ray powder diffraction (XRPD);

FIGS. 2A-2B provide a flowchart of steps for implementing the TriadsAlgorithm and associated procedures, where the steps of the TriadsAlgorithm are set out in the box in FIG. 2A;

FIG. 3 illustrates an XRPD pattern calculated from the CambridgeStructural Database structure ABIVIQ, using the wavelength of Cu—Kαradiation;

FIG. 4 illustrates the XRPD pattern of FIG. 3 with bars below the axisto indicate the positions of reflections calculated using Bragg's Lawand the reciprocal cell parameters generated using the Triads Algorithm;and

FIG. 5 illustrates the XRPD pattern of FIG. 3 with bars below the axisto indicate the positions of reflections for the conventional unit celland extinction symbol.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

Reference will now be made in greater detail to exemplary embodiments ofthe invention. It is to be understood that both the foregoing generaldescription and following detailed description are exemplary andexplanatory only, and are not to be interpreted as restrictive of theinvention as claimed. It will be apparent to one skilled in the art thatthe invention may be practiced without some or all of these specificdetails, and the disclosure is intended to cover alternatives,modifications, and equivalents. For example, well-known features and/orprocess steps may not have been described in detail so as not tounnecessarily obscure the invention.

This invention relates to the characterization of crystalline solidforms. The invention includes methods for determining the unit cellparameters of a crystalline solid form by using an indexing algorithmreferred to as the Triads Algorithm.

Crystalline solid forms may be generated in numerous ways. For example,samples may be crystallized in different environments by, for instance,using different solvents, different temperatures, different humidities,or different pressures. Such different conditions increase thelikelihood of obtaining more than one crystalline solid form of acompound.

An indexed unit cell can, in various embodiments, be used to determinethe relationship between different crystalline solid forms of a singlemolecule. For example, it can assist in determining whether thecrystalline solid forms are iso-structural and/or part of a singlehydrate family. Indexing can be used to rule out erroneous claims of newforms arising from, for example, poor particle statistics or preferredorientation artifacts. If an indexed crystal unit cell describes allmeasured diffraction peaks in a powder pattern, then most likely thesample material has the same crystal unit cell.

The indexing method of the invention may be applied, for example, todistinguish between different crystalline solid forms of a substance.This method may be used, for example, in a screen for identifying newcrystalline solid forms of a substance. The ability to index a measuredpowder pattern may also rule out the possibility that the samplematerial is a mixture of different crystalline solid forms. The inversecan also be true. If a powder pattern cannot be indexed, then the samplematerial may be a mixture of different crystalline solid forms, which isanother source of false form identification.

The Triads Algorithm allows for the identification of members of theinfinite set of crystallographic unit cells that describes a given XRPDpattern, without regard for whether or not those cells conform toconventions for their proper description. In other words, a convenient,rather than conventional, unit cell can be determined. In certainembodiments of the invention, known methods may be used to reduce anarbitrary unit cell to the reduced basis or for transforming the reducedbasis to the conventional unit cell. The Triads Algorithm may be moreuseful than other methods in certain applications, such as, for example,low symmetry crystal structures such as those commonly found formolecular solids.

In one embodiment, the method selects three sets of three peaks, calledTriads, in such a way as to generate unit cells that may describe aparticular XRPD pattern. Although some of the peaks may appear in morethan one triad, up to six peaks from the XRPD pattern may be used togenerate a candidate unit cell. Additional peaks may be needed to verifythe candidate unit cell.

In the following description, step numbers correspond to labels in FIGS.2A and 2B:

Step 1:

In a first step of an embodiment of the Triads Algorithm method of theinvention, an X-ray powder diffractometer may generate one or more XRPDpatterns of a crystalline solid form. FIG. 2A, 1. Examples of suchdiffractometers include any such instruments known in the art including,for example, the Siemens D-500 X-ray Powder Diffractometer-Kristalloflexand a Shimadzu XRD-6000 X-ray powder diffractometer. A common choice isto use Cu—Kα radiation, but Mo and other metal anodes may also be used.

Step 2:

Once XRPD patterns of the crystalline solid form are generated, a peaklist may then be prepared corresponding to the XPRD pattern. FIG. 2A, 2.Typically, the peak list indicates scattering angles (2θ[deg]) at whichhigher than background intensity is observed. In one embodiment,low-angle peaks, regardless of intensity, may be selected. The term“low-angle” is well known in the art and refers to peaks with values ofκ that are within the ranges specified by equation (5) below. Sincethese ranges may be dependent upon the selected basis peaks, it isdifficult to provide an upper limit on the range for low-angle peaks.For low-angle basis peaks (Step 4), the specified vector sum ranges arealso for relatively low-angle peaks. Inclusion of higher angle peaks inthe observed peak list does not interfere with the success of thealgorithm and may, in some embodiments, aid in solution refinement(Steps 10, 13, 16, and 18). Longer observed peak lists necessitatelonger calculated peak lists (Step 8) which may extend the totalexecution time for the algorithm. One of skill in the art will generallyappreciate which peaks are considered “low-angle” for any particularsample.

Step 3:

It is convenient to express the peak list as a function of sin(θ),κ[1/Å], κ²[1/Å²], and/or d/n [Å] where κ is the magnitude of thescattering vector and d/n is the distance between Miller planes dividedby the order of the reflection (See equation 2), and λ is the wavelengthof the incident X-ray radiation. For example, λ is 1.54059 Å for Cu—Kα₁radiation. FIG. 2A, 3.

Step 4:

In a further step of an embodiment of the Triads Algorithm method of theinvention, three peaks from the peak list are chosen and labeled as A,B, and C. FIG. 2A, 4. In the following description, these three peaksare called basis peaks. The basis peaks may be chosen in any order. Itis convenient to select the peaks in a way that avoids permutations ofthe labels since such permutations do not lead to independent indexingsolutions. For example, the peaks may be chosen such thatκ_(C)≦κ_(B)≦κ_(A).

Step 5:

Next, in a further step of an embodiment of the Triads Algorithm methodof the invention, peaks in the XRPD pattern, and thus the peak list, maybe determined for the vector sums of pairs of C, B, and A, within thefollowing ranges of equation (5):|κ_(B)−κ_(C)|≦κ_(B+C)≦κ_(B)+κ_(C)|κ_(A)−κ_(C)|≦κ_(A+C)≦κ_(A)+κ_(C)|κ_(A)−κ_(B)|≦κ_(A+B)≦κ_(A)+κ_(B)  (5)FIG. 2A, 5. The indicated ranges follow from the interpretation ofκ_(X+Y) as a vector sum of κ_(X) and κ_(Y), where X and Y are selectedfrom A, B, and C. Since the reciprocal angle cosines in equation (4) arebounded on −1 to +1, it follows that the magnitude of the vector sumpeaks are bounded by the sum and difference of the component vectors.Since B, C, and B+C appear together in equation (5), they are termed aTriad which is the origin of the name of the algorithm. Note that theabsolute magnitude indicated in equation (5) is redundant if the basispeaks are assigned as indicated in Step 4.Step 6:

In a further step of an embodiment of the Triads Algorithm method of theinvention, three peaks from the peak list within the ranges specified inStep 5 are chosen and labeled as B+C, A+C, and A+B. FIG. 2A, 6.

Step 7:

In a further step of an embodiment of the Triads Algorithm method of theinvention, the reciprocal cell length parameters (a*,b*,c*) can becalculated from κ_(C), κ_(B), and κ_(A) based on the assignment of theMiller index labels of (001), (010), and (100), respectively. FIG. 2A,7. This can be done by methods known to those of skill in the art. Notethat this portion of Step 7 may be accomplished following Step 4 above.This is efficient since the basis peaks remain fixed while the vectorsum peaks are repeatedly chosen and tested.

$\begin{matrix}{{\begin{bmatrix}\left( {1/{\Delta\kappa}_{A}^{2}} \right) & 0 & 0 \\0 & \left( {1/{\Delta\kappa}_{B}^{2}} \right) & 0 \\0 & 0 & \left( {1/{\Delta\kappa}_{C}^{2}} \right)\end{bmatrix} \cdot \begin{bmatrix}\left( a^{*} \right)^{2} \\\left( b^{*} \right)^{2} \\\left( c^{*} \right)^{2}\end{bmatrix}} = \begin{bmatrix}{\kappa_{A}^{2}/{\Delta\kappa}_{A}^{2}} \\{\kappa_{B}^{2}/{\Delta\kappa}_{B}^{2}} \\{\kappa_{C}^{2}/{\Delta\kappa}_{C}^{2}}\end{bmatrix}} & (6)\end{matrix}$

In a further step of an embodiment of the Triads Algorithm method of theinvention, the cosines of the reciprocal cell angle parameters(α*,β*,γ*) can be calculated from κ_(B+C), κ_(A+C), and κ_(A+B), underthe assumption that their Miller index labels are (011), (101), and(110), respectively.

$\begin{matrix}{{\begin{bmatrix}\left( \frac{2}{{\Delta\kappa}_{B + C}^{2}} \right) & 0 & 0 \\0 & \left( \frac{2}{{\Delta\kappa}_{A + C}^{2}} \right) & 0 \\0 & 0 & \left( \frac{2}{{\Delta\kappa}_{A + B}^{2}} \right)\end{bmatrix} \cdot \begin{bmatrix}{b^{*}c^{*}{\cos\left( \alpha^{*} \right)}} \\{a^{*}c^{*}{\cos\left( \beta^{*} \right)}} \\{a^{*}b^{*}{\cos\left( \gamma^{*} \right)}}\end{bmatrix}} = {\quad{\begin{bmatrix}\left( \frac{\kappa_{B + C}^{2}}{{\Delta\kappa}_{B + C}^{2}} \right) \\\left( \frac{\kappa_{A + C}^{2}}{{\Delta\kappa}_{A + C}^{2}} \right) \\\left( \frac{\kappa_{A + B}^{2}}{{\Delta\kappa}_{A + B}^{2}} \right)\end{bmatrix}{\quad{- {\quad{\begin{bmatrix}0 & \left( \frac{1}{{\Delta\kappa}_{B + C}^{2}} \right) & \left( \frac{1}{{\Delta\kappa}_{B + C}^{2}} \right) \\\left( \frac{1}{{\Delta\kappa}_{A + C}^{2}} \right) & 0 & \left( \frac{1}{{\Delta\kappa}_{A + C}^{2}} \right) \\\left( \frac{1}{{\Delta\kappa}_{A + B}^{2}} \right) & \left( \frac{1}{{\Delta\kappa}_{A + B}^{2}} \right) & 0\end{bmatrix}{\quad{\cdot \begin{bmatrix}\left( a^{*} \right)^{2} \\\left( b^{*} \right)^{2} \\\left( c^{*} \right)^{2}\end{bmatrix}}}}}}}}}} & (7)\end{matrix}$

Assignment of convenient Miller indices to selected peaks is permissiblein the Triads Algorithm because of the non-uniqueness ofcrystallographic unit cells. By assigning the Miller index label (100)to peak A, the Algorithm is selecting a subset from the infinite set ofequivalent unit cells. Assignment of Miller indices to the other peakssimilarly selects subsets of equivalent unit cells. The intersection ofthese sets may be a null set indicating that new choices of basis peaks(A, B, and C) and/or vector sum peaks (B+C, A+C, and A+B) are needed.For some choices of basis peaks and vector sum peaks, the TriadsAlgorithm generates candidate unit cells. By construction, those cellsare consistent with the six selected basis and vector sum peaks. Thecells may then be evaluated to assess their consistency with theremaining peaks in the peak list.

Step 8:

The candidate unit cell generated using the Triads Algorithm may then beevaluated to determine if the cell is consistent with the entire peaklist. FIG. 2A, 8. The first step is to calculate a peak list from theunit cell. This may be done, for example, by constructing a Bragg Matrix(B) from the calculated reciprocal cell parameters generated via theTriads Algorithm, as in equation (4), and then carrying out theindicated matrix multiplications in equation (3) for a series of Millerindex triples (hkl). Sufficient ranges of the Miller indices areestimated from the reciprocal cell parameters. For the Miller index h,the maximum requisite value is given by:

$\begin{matrix}{h^{\max} = \left\lfloor \frac{\kappa^{\max}{\sin\left( \alpha^{*} \right)}}{{\alpha^{*}\begin{pmatrix}{1 - {\cos^{2}\left( \alpha^{*} \right)} - {\cos^{2}\left( \beta^{*} \right)} - {\cos^{2}\left( \gamma^{*} \right)} +} \\{2{\cos\left( \alpha^{*} \right)}{\cos\left( \beta^{*} \right)}{\cos\left( \gamma^{*} \right)}}\end{pmatrix}}^{\frac{1}{2}}} \right\rfloor} & (8)\end{matrix}$where └f┘ denotes the floor function applied to f, and κ^(max) is themaximum value of κ for the peak list. Equation (8) may be expressed moresimply using real cell parameters instead of reciprocal cell parameters:h ^(max) =└aκ ^(max)┘  (9)Analogous equations for k^(max) and l^(max) can be obtained using thefollowing substitutions:

-   To calculate k^(max),

Symbol in equation (8) Replacement a* b* b* a* c* unchanged α* β* β* α*γ* unchangedor using real cell parameters:k ^(max) =└bκ ^(max)┘.  (10)

-   To calculate l^(max),

Symbol in equation (8) Replacement a* c* b* unchanged c* a* α* γ* β*unchanged γ* α*or using real cell parameters:l ^(max) =└cκ ^(max)┘.  (11)The relevant ranges for (h,k,l) are −h^(max)≦h≦h^(max), and similarlyfor k and l. Note the comments regarding Friedel's Law followingequation (4) above may be used to further limit the sufficient domainsin keeping with a particular choice of naming conventions.Step 9:

In various exemplary embodiments, all of the observed peaks will beindexed by the candidate unit cell with appropriate choices of Millerindices. FIG. 2A, 9. In a few cases there are extra allowed reflectionsthat are either extinct or just very weak, but all of the observed peaksmay, for example, correspond to one or more Miller index triplets. Inexemplary embodiments where all observed peaks are indexed by the trialunit cell and there are relatively few extra peaks, the trial unit cellis considered further. In exemplary embodiments where the agreement isunsatisfactory, then alternative choices for labeled peaks (A, B, C,A+B, A+C, and B+C) would be generated and evaluated.

Step 10:

The Triads Indexing Algorithm is well suited for unit cell parameterrefinement. FIG. 2A, 10. The three basis peaks (A, B, and C) and threevector sum peaks (B+C, A+C, and A+B) form a linearly independent basisfor determining the six reciprocal cell parameters as demonstrated inequations (6) and (7) of Step 7. For refinement, these six peaks areaugmented with observed peaks whose positions are consistent with oneand only one calculated peak, within their combined uncertainties. Suchpeaks are called Uniquely Indexed Peaks (UIPs). The union of the sets ofbasis peaks, vector sum peaks, and UIPs form a set of n peaks to be usedin the refinement step. The coefficient matrix A, unknown vector x, andconstant vector b are defined as follows:

$\begin{matrix}{\mspace{79mu}{\underset{\underset{\_}{\_}}{A} = \begin{bmatrix}\left( \frac{h_{1}^{2}}{{\Delta\kappa}_{1}^{2}} \right) & \left( \frac{k_{1}^{2}}{{\Delta\kappa}_{1}^{2}} \right) & \left( \frac{l_{1}^{2}}{{\Delta\kappa}_{1}^{2}} \right) & \left( \frac{2k_{1}l_{1}}{{\Delta\kappa}_{1}^{2}} \right) & \left( \frac{2h_{1}l_{1}}{{\Delta\kappa}_{1}^{2}} \right) & \left( \frac{2h_{1}k_{1}}{{\Delta\kappa}_{1}^{2}} \right) \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\\left( \frac{h_{n}^{2}}{{\Delta\kappa}_{n}^{2}} \right) & \left( \frac{k_{n}^{2}}{{\Delta\kappa}_{n}^{2}} \right) & \left( \frac{l_{n}^{2}}{{\Delta\kappa}_{n}^{2}} \right) & \left( \frac{2k_{n}l_{n}}{{\Delta\kappa}_{n}^{2}} \right) & \left( \frac{2h_{n}l_{n}}{{\Delta\kappa}_{n}^{2}} \right) & \left( \frac{2h_{n}k_{n}}{{\Delta\kappa}_{n}^{2}} \right)\end{bmatrix}}} & (12) \\{{\underset{\_}{x}}^{t} = {\quad\begin{bmatrix}\left( a^{*} \right)^{2} & \left( b^{*} \right)^{2} & \left( c^{*} \right)^{2} & {b^{*}c^{*}{\cos\left( a^{*} \right)}^{2}} & {\;{a^{*}c^{*}{\cos\left( \beta^{*} \right)}^{2}}} & {a^{*}b^{*}{\cos\left( \gamma^{*} \right)}^{2}}\end{bmatrix}}} & (13) \\{\mspace{79mu}{{\underset{\_}{b}}^{t} = \begin{bmatrix}\left( \frac{\kappa_{1}^{2}}{{\Delta\kappa}_{1}^{2}} \right) & \cdots & \left( \frac{\kappa_{n}^{2}}{{\Delta\kappa}_{n}^{2}} \right)\end{bmatrix}}} & (14) \\{\mspace{79mu}{{\underset{\underset{\_}{\_}}{A} \cdot \underset{\_}{x}} \approx \underset{\_}{b}}} & (15)\end{matrix}$Note that there are at least six rows in A and b corresponding to thebasis and vector sum peaks. UIPs, if not already included, addadditional rows. There are precisely six columns in A and six rows in xcorresponding to the six unknown reciprocal lattice parameters. The ‘≈’sign in equation (15) recognizes that the system is over specified ifUIPs have been used to augment A and b. Therefore, the solution for x isunderstood to be the optimal solution in a least-squares sense.

Note that equations (6) and (7) are special cases of equations (12)through (15).

Solution of equation (15) is readily accomplished using the “NormalEquation Method.” The Covariance matrix (C) is calculated from thecoefficient matrix (A) as follows:C =( A ^(t) ·A )⁻¹  (16)then the best solution for x is given by:x≈C·A ^(t) ·b   (17)The ‘≈’ sign in equation (17) is in recognition that the solution for xis a least-squares estimate.

Once the vector (x) is known, the position of any Bragg reflection withMiller Indices (h_(i),k_(i),l_(i)) is estimated by:v _(i) ^(t) =[h _(i) ² k _(i) ² l _(i) ²2k _(i) l _(i)2h _(i) l_(i)2h_(i) k _(i)]  (18)κ_(i) ² ≈v _(i) ^(t) ·x   (19)and the uncertainty in the Bragg position is:Δκ_(i) ²≈( v _(i) ^(t) ·C˜v _(i))^(1/2)  (20)Expanding equations (3) and (19) proves that they are equivalent.

Augmenting the Triads peaks with uniquely indexed peaks (if any) oftenresults in smaller uncertainties in the Bragg positions. In turn, thisleads to additional UIPs. Thus the peak list comparison (Step 9) andrefinement (Step 10) may be applied iteratively until the list of UIPsis unchanged upon successive applications of the refinement process orthe observed and calculated peak lists are found to be inconsistent.

Step 11:

In a further step of an exemplary embodiment of the invention, realspace cell parameters may then be calculated. FIG. 2B, 11. This may bedone by any method known to those of skill in the art.

Step 12:

In a further step of an exemplary embodiment, the reduced basis may thenbe calculated, as the real unit cell generally does not correspond tostandard conventions. FIG. 2B, 12. This may be done by any method knownto those of skill in the art.

Although the reduced basis is unique, it may not immediately convey thesymmetry of many unit cells. Thus, in a further step of variousadditional embodiments of the invention, the conventional unit cell maybe calculated. Conventional unit cells are constructed to convey theunderlying cell symmetry more readily. To construct the conventionalunit cell, the character number for the reduced basis is identified andthe indicated cell transformation is applied. Methods for doing so areknown to those of skill in the art.

Step 13:

Using the reduced basis, the low-angle Bragg reflections are labeledwith small Miller indices. This is often not the case for the unit cellprior to the cell reduction (Step 12). Since the Miller indices (h,k,l)for individual reflections change upon reduction, the correspondingcoefficient matrix A in equation (12) and covariance matrix C inequation (16) also change. As a result, the calculated uncertainty for aparticular peak, Δκ², changes as a result of unit cell reduction.Modified peak position estimates may lead to additional UIPs whichjustifies another round of refinement. The process is the same asdescribed in Step 10, except that the Miller indices must refer to thereduced cell basis.

If the unit cell prior to the cell reduction has a matrix representationM and the transformation matrix for the cell reduction is T, then thematrix representation of the reduced basis is given by:M _(reduced) =M·T   (21)The same transformation matrix converts the Miller indices from theoriginal basis to the reduced basis:

$\begin{matrix}{\begin{bmatrix}h \\k \\l\end{bmatrix}_{reduced} = {{\underset{\underset{\_}{\_}}{T}}^{t} \cdot \begin{bmatrix}h \\k \\l\end{bmatrix}}} & (22)\end{matrix}$The Miller indices (h,k,l) in the reduced basis are used in therefinement process. FIG. 2B, 13.Step 14:

Once the reduced basis has been determined, the character correspondingto the reduced basis is determined by comparison with tabulateddescriptions. International Tables for Crystallography, Vol A, 5^(th)ed., §9.2, pp. 750-755. FIG. 2B, 14.

Step 15:

For each character there is a corresponding matrix transformation thatconverts the reduced basis to the conventional unit cell. FIG. 2B, 15.Application of the matrix transformation is similar to that in equation(21).

Step 16:

Following conversion to the conventional unit cell, the Miller indicesfor individual reflections must be converted to the conventional unitcell basis. Transforming the Miller indices from the reduced basis tothe conventional basis is similar to that in equation (22). For similarreasons to those given in Step 13, this justifies another round of cellrefinement which uses the conventional cell Miller indices, but isotherwise similar to that described in Step 10. FIG. 2B, 16.

Step 17:

In further exemplary embodiments of the invention, additional processingsteps may also be performed. By way of non-limiting example, in certaincases the extinction symbol may be applied such as when systemicextinctions due to additional symmetry of the unit cell contents exist.FIG. 2B, 17. In those cases, applying the extinction symbol may giveshorter calculated peak lists that match the observed peak list withfewer unobserved peaks. In these cases the extinction symbol indicatesthe presence of glide planes and/or screw axes in the crystal structure.Note that cell centering, which also yields systematic extinctions, hadalready been recognized in Step 15 as part of the conventional unit cellconstruction.

Step 18:

Since application of extinction symbols may eliminate reflections thathad interfered with the identification of UIPs, a final round ofrefinement may be warranted. FIG. 2B, 18.

Step 19:

Most trial solutions do not reach step 19 since they are found to beinconsistent in Step 9 or in one of the refinement steps (Steps 10, 13,16, or 18). The trial solutions that do survive to Step 19 areconsistent with the observed peak list, but not all consistent trialsolutions are optimal. Solutions with fewer calculated peaks, moreuniquely indexed peaks, and smaller peak uncertainties are oftenconsidered preferred solutions than more complex and/or less accuratesolutions, and thus are favorable. Therefore, trial solutions reachingStep 19 are evaluated, compared with previously generated trialsolutions, and recorded if the solutions are judged to be favorable.FIG. 2B, 19.

Step 20:

As discussed above, indexing results can be used to distinguish orscreen crystalline solid forms such as pharmaceuticals. Therefore, anadditional exemplary embodiment of the invention comprises comparingstructural information obtained for different crystalline solid samples,such as the indexed unit cell, to determine whether XRPD patterns ofthose samples represent the same or different crystalline solid forms.FIGS. 2A-2B, 20.

This embodiment can comprise, for example, comparing structuralinformation obtained for different crystalline solid samples, such asthe results obtained from the method of the invention, to determinewhether XRPD patterns of those samples represent the same or differentcrystalline solid forms. The calculation of the same crystal unit cellparameters can indicate that the samples have the same crystalline solidform. Conversely, the calculation of different crystal unit cellparameters for samples represented by different XRPD patterns canindicate that the samples do not have the same crystalline solid form.One of skill in the art, evaluating the results of the methods describedherein, can determine whether two samples represent the same ordifferent forms, based on knowledge and techniques that are well known.

Another embodiment of the invention comprises sorting, screening, orranking various crystalline solid forms on the basis of certainstructural information specific to the forms, such as, for example, theunit cell parameters, character, and/or extinction symbol of eachcrystalline solid form. For instance, the invention comprises methods ofscreening for new crystalline solid forms of a substance, whichcomprises determining structural information for a plurality ofcrystalline samples of a substance using the embodiments describedabove, comparing the structural information of the samples to structuralinformation of known crystalline solid forms of the substances, andidentifying those crystalline samples that have structural informationdifferent from that of the known crystalline solid forms.

It should be noted that the terms “same” or “similar,” as used herein,such as, for example, when referring to unit cell parameters, are meantto convey that a particular set of data is, within acceptable scientificlimits, sufficiently similar to another such that one of skill in theart would appreciate that the data represent, for example, the samecrystalline solid form of a compound. In this regard, and as is commonlypracticed within the scientific community, it is not intended that thedata be identical in order to be considered the same or similar.

The present invention is further illustrated by the followingnon-limiting examples, which are provided to further aid those of skillin the art in the appreciation of the invention.

It is noted that, as used in this specification and the appended claims,the singular forms “a,” “an,” and “the,” include plural referents unlessexpressly and unequivocally limited to one referent, and vice versa.Thus, by way of example only, reference to “a crystalline solid form”can refer to one or more crystalline solid forms, and reference to “acompound” can refer to one or more compounds. As used herein, the term“include” and its grammatical variants are intended to be non-limiting,for example such that recitation of items in a list is not to theexclusion of other like items that can be substituted or added to thelisted items.

It will be apparent to those skilled in the art that variousmodifications and variations can be made to the present disclosurewithout departing from the scope of its teachings. Other embodiments ofthe disclosure will be apparent to those skilled in the art fromconsideration of the specification and practice of the teachingsdisclosed herein, such as, for example, the use of the methods hereinwith powder diffraction data other than XRPD data. It is intended thatthe embodiments described in the specification be considered asexemplary only. Additional objects and advantages of the invention areset forth in the following description. For instance, it will be notedthat the order of the steps presented need not necessarily be performedin that order set forth herein to practice the invention, and some stepsmay be changed or omitted all together.

EXAMPLES Example 1

In the following example, which is not intended to be limiting of theinvention as claimed, the Triads Algorithm is applied to a crystallinematerial whose structure is stored in the Cambridge Structural Database(CSD), with the ABIVIQ reference code. The corresponding molecule is1,4-Dimethyl-2,5-dioxabicyclo(2.2.1)heptane-3,6-dione. Its crystalstructure was previously determined. In other words, Example 1 usessimulated data obtained from the CSD, rather thanexperimentally-obtained data, in order to demonstrate the accuracy andprecision of the methods of the invention.

Step 1:

As an initial step, the XRPD pattern for ABIVIQ was calculated toapproximate an experimental XRPD pattern measured using Cu—Kα radiation.The pattern was calculated using the MERCURY software package. Thecalculated XRPD pattern is presented in FIG. 3.

Steps 2-4:

Next, the following peak list of TABLE 2 was generated by picking peaksfrom the XRPD pattern of FIG. 3 and calculating corresponding values forsin(θ) and the scattering vector magnitude (κ) for each peak. Threepeaks were selected and labeled as A, B, and C. The indicated peaks wereselected for the example because they lead to a correct indexingsolution. Most choices for basis peaks A, B, and C do not lead tocorrect indexing solutions, but the illustrated choice is not uniqueeither.

TABLE 2 Peak list for ABIVIQ generated from XRPD pattern of FIG. 3 withbasis peaks (A, B, and C) labeled. 2θ [deg] sin(θ) κ [1/Å] Label 13.620.1186 0.1539 C 15.12 0.1316 0.1708 B 18.74 0.1628 0.2114 A 19.84 0.17230.2236 20.16 0.1750 0.2272 22.96 0.1990 0.2584 23.20 0.2011 0.2610 27.420.2370 0.3077 30.02 0.2590 0.3362 30.54 0.2634 0.3419 31.44 0.27090.3517 32.14 0.2768 0.3594 32.74 0.2818 0.3659 33.58 0.2889 0.3750 34.900.2999 0.3893Step 7 (Partial):

Next, the following reciprocal cell length parameters (a*,b*,c*) werecalculated from κ_(C), κ_(B), and κ_(A) on the assumption that theirMiller index labels are (001), (010), and (100), respectively:c*=κ _(C)=0.1539/Åb*=κ _(B)=0.1708/Åa*=κ _(A)=0.2114/ÅStep 5:

Subsequently, peaks for the vector sums of pairs C, B, and A weredetermined within the ranges of equation (5). For the current example,κ_(B)=0.1708/Å and κ_(C)=0.1539/Å. Therefore:0.0169/Å=κ_(B)−κ_(C)≦κ_(B+C)≦κ_(B)+κ_(C)=0.3247/ÅThere are eight candidate peaks within the specified range in the XRPDpattern of FIG. 2. Similarly, since κ_(A)=0.2114/Å:0.0575/Å=κ_(A)−κ_(C)≦κ_(A+C)≦κ_(A)+κ_(C)=0.3653/Å0.0406/Å=κ_(A)−κ_(B)≦κ_(A+B)≦κ_(A)+κ_(B)=0.3822/ÅThere are twelve candidate peaks within the specified range for A+C andfourteen candidate peaks within the specified range for A+B.Step 6:

Accordingly, there are 8*12*14=1344 combinations of triads of the form{κ_(B),κ_(C),κ_(B+C)}, {κ_(A),κ_(C),κ_(A+C)}, {κ_(A),κ_(B),κ_(A+B)} forthe choice of basis peaks (A, B, and C) and the peak list provided inTABLE 2. One such combination is indicated in TABLE 3:

TABLE 3 Peak list for ABIVIQ from TABLE 2 with vector sum peaks (B + C,A + C, and A + B) labeled. 2θ [deg] sin(θ) κ [1/Å] Label 13.62 0.11860.1539 C 15.12 0.1316 0.1708 B 18.74 0.1628 0.2114 A 19.84 0.1723 0.223620.16 0.1750 0.2272 A + B 22.96 0.1990 0.2584 B + C 23.20 0.2011 0.261027.42 0.2370 0.3077 30.02 0.2590 0.3362 A + C 30.54 0.2634 0.3419 31.440.2709 0.3517 32.14 0.2768 0.3594 32.74 0.2818 0.3659 33.58 0.28890.3750 34.90 0.2999 0.3893Step 7 (Continued):

Further, the cosines of the reciprocal cell angle parameters (α*,β*,γ*)were calculated from κ_(B+C), κ_(A+C), and κ_(A+B) with the assignedMiller index labels (011), (101), and (110), respectively:

${\cos\left( a^{*} \right)} = {\frac{\left( \kappa_{B + C} \right)^{2} - \left( b^{*} \right)^{2} - \left( c^{*} \right)^{2}}{2\; b^{*}c^{*}} = 0.2641}$${\cos\left( \beta^{*} \right)} = {\frac{\left( \kappa_{A + C} \right)^{2} - \left( a^{*} \right)^{2} - \left( c^{*} \right)^{2}}{2\; a^{*}c^{*}} = 0.6865}$${\cos\left( \gamma^{*} \right)} = {\frac{\left( \kappa_{A + B} \right)^{2} - \left( a^{*} \right)^{2} - \left( b^{*} \right)^{2}}{2\; a^{*}b^{*}} = {- 0.3077}}$

TABLE 4 Reciprocal space unit cell parameters calculated for ABIVIQ(initial cell). a* = 0.2114 Å⁻¹ b* = 0.1708 Å⁻¹ c* = 0.1539 Å⁻¹ cos(α*)= 0.2641 cos(β*) = 0.6865 cos(γ*) = −0.3077  V* = 0.002793 Å⁻³ The reciprocal cell volume (V*) is similar in magnitude to(k₁)³=0.003648/Å³. If the reciprocal cell volume had been much smallerthan (k₁)³ then the trial solution would be rejected since such a cellwould have too many calculated peaks to be judged acceptable in Step 19.Since the trial cell reciprocal volume and the estimate are similar inmagnitude, the algorithm continues with Step 8.Step 8:

Next, the trial unit cell was evaluated by constructing the Bragg Matrix(B) from the calculated reciprocal cell parameters and equations (3) and(4). FIG. 4 shows that all of the observed peaks are indexed by thecandidate unit cell with appropriate choices of Miller indices. The barsbelow the axis indicate the positions of reflections calculated usingBragg's Law and the reciprocal cell parameters generated using theTriads Algorithm. In a few cases there are extra allowed reflectionsthat are not evident, either because they are extinct or just very weak,but all of the observed peaks correspond to one or more Miller indextriplets. Since all observed peaks were indexed by the trial unit celland there are relatively few extra peaks, the trial unit cell is furtherevaluated. If the agreement had been unsatisfactory, then alternativechoices for labeled peaks (A, B, C, A+B, A+C, and B+C in TABLE 2) couldhave been generated and evaluated.

Step 9:

Since there is at least one calculated peak position (Step 8) for eachobserved peak (Step 2) then the peak lists are judged to match and thealgorithm continues with Step 10 below.

Step 10:

For the current example and at this stage of the algorithm, the only UIPis also the A basis peak. Since there are no additional peaks beyondthose already used in the calculation of the reciprocal latticeparameters, there is no benefit from the refinement step. Therefore,this refinement step is skipped.

Step 11:

Next, the real space parameters were calculated using methods known inthe art.

TABLE 5 Real space unit cell parameters calculated for ABIVIQ(unconventional cell). a = 9.079 Å b = 8.470 Å c = 12.298 Å  α = 133.41°β = 146.79° γ =  45.78°Step 12:

Subsequently, the reduced basis was calculated using methods known inthe art.

TABLE 6 Reduced basis real parameters calculated for ABIVIQ. a = 6.843 Åb = 6.849 Å c = 8.470 Å α = 71.81° β = 71.96° γ = 81.46°Despite the change in lattice parameters and Miller index labels, thepeak positions illustrated in FIG. 4 are unchanged by the basisreduction procedure.Step 13:

For the current example and at this stage of the algorithm, there aretwo UIPs: one basis peak and one other. The second UIP facilitatesrefinement of the reciprocal lattice parameters of the reduced basis. Itis used to augment the coefficient matrix (A) and the vector (b). Theresulting reciprocal lattice parameters are:

TABLE 7 Refined reduced basis reciprocal parameters calculated forABIVIQ. a* = 0.1539 Å⁻¹ b* = 0.1538 Å⁻¹ c* = 0.1293 Å⁻¹ cos(α*) =−0.2816 cos(β*) = −0.2819 cos(γ*) = −0.0565 V* = 0.002787 Å⁻³ Step 14:

Converting the reciprocal parameters in TABLE 7 to real space, using thesame method as in Step 11, and then comparing with tabulated characterdescriptions leads to the conclusion that the reduced basis conforms tocharacter 10, within reasonable round-off errors.

Step 15:

This allows the conventional unit cell to be determined by applying theindicated cell transformation for character 10, which generates acentered monoclinic conventional unit cell with the following parametersusing methods known in the art. The requisite matrix transformations aretabulated and the procedure for applying the matrix transformation isdocumented elsewhere.

TABLE 8 Conventional cell parameters calculated for ABIVIQ. CellC-Centered a = 10.378 Å  b = 8.944 Å c = 8.478 Å α = 90.00° β = 114.25° γ = 90.00°Despite the change in parameters and Miller index labels, the allowedpeak positions illustrated in FIG. 4 are unchanged. The parameterslisted in TABLE 8 conform to the usual monoclinic conventional celldefinition (with α=γ=90°) which is not evident in the reduced basis ofTABLE 6.Step 16:

Optional refinement using reciprocal lattice parameters for conventionalcell leads to the slightly different conventional cell parameters listedin TABLE 9.

TABLE 9 Refined conventional cell parameters calculated for ABIVIQ. CellC-Centered a = 10.376 Å  b = 8.944 Å c = 8.477 Å α = 90.00° β = 114.24° γ = 90.00°Step 17:

Some of the allowed peak positions labeled with bars below the axis inFIG. 4 do not correspond to observed peaks. This may be the result ofpeaks that are too weak or may be due to systematic extinctions due toadditional symmetry of the unit cell contents. An extinction symbol wassought using methods known to those of skill in the art to account foras many of these unobserved peaks as possible while maintainingconsistency with the observed peaks. Such techniques can be applied todetermine that the systematic extinctions are consistent with themonoclinic C1c1 extinction symbol. Applying that extinction symbol givesthe allowed peak positions illustrated in FIG. 5. The excellentagreement in peak positions with a lack of unobserved peaks indicatesthat the conventional unit cell and extinction symbol are an acceptableindexing solution for the XRPD pattern.

Step 18:

Optional refinement using reciprocal lattice parameters for theconventional cell and applied extinction symbol leads to the valuestabulated in the leftmost column of TABLE 10. The middle column containsthe conventional cell parameters in the CSD entry used to construct theXRPD pattern in Step 1. The rightmost column contains their difference.

TABLE 10 Refined conventional cell parameters calculated for ABIVIQ withextinction symbol determination. Triads Algorithm CSD: ABIVIQ DifferenceExtinction Symbol C1c1 C1c1 n/a a = 10.376 Å  10.379 Å  −0.003 Å b =8.944 Å 8.943 Å −0.001 Å c = 8.477 Å 8.477 Å  0.000 Å α = 90.00° 90° 0°β = 114.24°   114.22°  −0.02° γ = 90.00° 90° 0° V = 717.4 Å³ 717.57 Å³  0.2 Å³Step 19:

TABLE 11 summarizes the ten highest ranked solutions generated with theTriads indexing algorithm and various choices of basis and vector sumpeaks. The table compares them to the known unit cell parameters forABIVIQ. The top row is the trial solution detailed above.

TABLE 11 Ten equivalent unit cell determinations for ABIVIQ comparedwith the known solution. A B C B + C A + C A + B a [Å] b [Å] c [Å] α[deg] β [deg] γ [deg] V [A³] 3 2 1 6 9 5 10.3762 8.9445 8.4775 90114.237 90 717.4 4 2 1 6 1 2 10.3764 8.9444 8.4776 90 114.239 90 717.4 42 1 6 11 12 10.3769 8.9444 8.4775 90 114.239 90 717.5 5 1 1 4 6 1310.3759 8.9443 8.4775 90 114.236 90 717.4 5 2 1 6 13 10 10.3760 8.94458.4777 90 114.233 90 717.5 5 3 1 1 13 2 10.3754 8.9446 8.4775 90 114.23390 717.4 5 3 1 9 6 2 10.3762 8.9445 8.4775 90 114.237 90 717.4 5 4 1 1 65 10.3759 8.9443 8.4775 90 114.235 90 717.4 5 4 1 1 13 5 10.3754 8.94468.4775 90 114.233 90 717.4 5 4 1 11 6 15 10.3769 8.9444 8.4775 90114.239 90 717.5 Mean 10.3761 8.9445 8.4775 90 114.236 90 717.43 ABIVIQ10.379 8.943 8.477 90 114.22 90 717.57 Difference −0.0029 0.0015 0.00050 0.016 0 −0.14 % diff −0.03% 0.02% 0.01% n/a 0.01% n/a −0.02%Step 20:

TABLE 11 demonstrates that the algorithm has yielded duplicate, highlyranked solutions which are in agreement with the known correct solution.While such duplicates are not necessary, they are indicative of asuccessful indexing solution. Successful indexing solution providesevidence of a single crystalline phase and a concise description of thepeak positions in its XRPD pattern.

Example 2

In the following example, which is not intended to be limiting of theinvention as claimed, the Triads Algorithm is applied to a crystallinesample of mannitol, form beta. Example 2 uses experimentally-obtaineddata.

A XRPD pattern was acquired using a PANalytical X'Pert Prodiffractometer. An incident beam of Cu Kα radiation was produced usingan Optix long, fine-focus source. An elliptically graded multilayermirror was used to focus the Cu Kα X-rays of the source through thespecimen and onto the detector. Data were collected and analysed usingX'Pert Pro Data Collector software (v. 2.2b). Prior to the analysis, asilicon specimen (NIST SRM 640c) was analyzed to verify the instrumentalignment using the Si 111 peak position. The specimen was sandwichedbetween 3 μm thick films, analyzed in transmission geometry, and rotatedto optimize orientation statistics. A beam-stop was used to minimize thebackground generated by air scattering. Soller slits were used for theincident and diffracted beams to minimize axial divergence. Diffractionpatterns were collected using a scanning position-sensitive detector(X'Celerator) located 240 mm from the specimen.

A peak list was constructed with 35 observed peaks and the TriadsAlgorithm was carried out. The results are given in TABLE 12. Excellentagreement between the Triads Algorithm results and previously reportedindexing results demonstrates the ability of the algorithm tosuccessfully index experimental data, in addition to the simulated XRPDpattern used in Examples 1 and 3. The differences in lattice parametersare slightly larger in Example 2 than in Examples 1 and 3, probably dueto differences in the samples and/or conditions of the analyzed samples.It is because of this uncertainty that a simulated XRPD pattern was usedto assess the accuracy and precision of the algorithm in Examples 1 and3.

TABLE 12 Two representative indexing solutions for Mannitol, form betacompared with a published structure. a [Å] b [Å] c [Å] α [deg] β [deg] γ[deg] V[Å³] Space Group 5.5491 8.6768 16.9039 90 90 90 813.90 P2₁2₁2₁(#19) 5.5491 8.6768 16.9039 90 90 90 813.90 P2₁2₁2₁ (#19) Mean 5.54918.6768 16.9039 90 90 90 813.896 n/a DMANTL04¹ 5.549 8.672 16.890 90 9090 812.762 P2₁2₁2₁ (#19) Difference 0.0001 0.0048 0.0138 0 0 0 1.134 n/a% diff 0.0018 0.055 0.082 n/a n/a n/a 0.14 n/a ¹The lattice parametersare sorted according to reduced cell convention for comparison.

Example 3

In the following example, which is not intended to be limiting of theinvention as claimed, the Triads Algorithm is applied to a set of XRPDpatterns collected from the CSD. Example 3 uses simulated data, whichdemonstrates that the methods of the invention are broadly applicable toall of the 44 reduced bases or characters.

The set includes representatives of each of the 44 characters with a fewduplicates. Characters 20 and 31 contain duplicates for a total of 46structures. The algorithm accurately identified 44 of 46 indexingsolutions. For AABHTZ and ABACUC, the indexing solutions were assignedto centered monoclinic rather than triclinic solutions. This is aproblem common to all indexing routines since indexing routinesdetermine metric symmetry and are therefore insensitive to the symmetryof the cell contents. In both of the problematic structures, the reducedcell (Step 12) provides the correct triclinic unit cell. Packing of themolecular contents would be necessary to correctly identify thetriclinic unit cell as the correct indexing solution in these cases. Forthe other 44 indexing solutions, the rms fractional error in unit celllengths was 0.022% and the rms error in unit cell angles was 0.0090degrees. Thus, the indexing of the invention provided agreement to thesource structure unit cell parameters with excellent precision andaccuracy. The small residual errors are the result of rounding errors inthe peak positions selected in Step 2.

TABLE 13 List of CSD structures used to test the Triads IndexingAlgorithm. Char. # Bravais Type CSD ID SG Ext. Symb 1 c F GIGRIX Fm-3m255 F - - - 2 h R AZTPHZ01 R-3 148 R - - 3 c P BZCBNL05 Pa-3 205 P a - -4 h R ASXANT01 R-3 148 R - - 5 c I BIGLUZ I-43m 217 I - - - 6 t I BAYJIUI41/a 88 I 41/a - - 7 t I ACAVIJ I41/a 88 I 41/a - - 8 o I FABGOE Iba245 I c - a 9 h R FIDJOS R-3 148 R - - 10 m C ABIVIQ C2/c 15 C 1 c 1 11 tP ACNORT P41212 92 P 41 21 - 12 h P BODROB P63/m 176 P 63 - - 13 o CBERCUW C2221 20 C - - 21 14 m C ABUBEE C2/c 15 C 1 c 1 15 t I GOLCALI41/a 88 I 41/a - - 16 o F ALUVAE F2dd 43 F - d d 17 m C ABIXUF C2/c 15C 1 c 1 18 t I AGURAV I41/a 88 I 41/a - - 19 o I ARIRAU Iba2 45 I c - a20 m C ABUNAM C2/c 15 C 1 c 1 20 m C ACXHTZ C2/c 15 C 1 c 1 21 t PCAPINC P41212 92 P 41 21 - 22 h P BARBOL P63/m 176 P 63 - - 23 o CTASKON C2221 20 C - - 21 24 h R BELQOZ R-3 148 R - - 25 m C ABOMEJ C2/c15 C 1 c 1 26 o F ARAGUV Fdd2 43 F - d d 27 m C ABIHOI C2/c 15 C 1 c 128 m C BAPMAH C2/c 15 C 1 c 1 29 m C AMAQUA C2/c 15 C 1 c 1 30 m CBESCIM C2/c 15 C 1 c 1 31 a P AABHTZ P-1 2 P - 31 a P ABACAI P-1 2 P -32 o P ABELAV P212121 19 P 21 21 21 33 m P POVJIT P21/c 14 P 1 21/c 1 34m P POXHDO P21/n 14 P 1 21/n 1 35 m P POVKEQ P21/n 14 P 1 21/n 1 36 o CPFBIPH01 C2221 20 C - - 21 37 m C ACEGUL C2/c 15 C 1 c 1 38 o C WAVYAUC2221 20 C - - 21 39 m C ACIBIX C2/c 15 C 1 c 1 40 o C LABTAK C2221 20C - - 21 41 m C AKADIZ C2/c 15 C 1 c 1 42 o I CIHCUR01 Iba2 45 I c - a43 m I BAWPOE I2/c 15 I 1 a 1 44 a P ABACUC P-1 2 P -

What is claimed is:
 1. A method for determining the unit cell parametersof a crystalline solid form of a compound, which method comprises: i.using a diffractometer to generate powder diffraction data for a solidcrystalline substance; and ii. determining the unit cell parameters ofthe substance by performing a Triads Algorithm to identify one or moresets of values of unit cell parameters of the crystalline solid form. 2.A method as in claim 1, which further comprises performing one or morerefinement steps.
 3. A method as in claim 1, wherein the powderdiffraction data is processed by the following step after it is obtainedand prior to the application of the Triads Algorithm: a. generating apeak list from the powder diffraction data of the crystalline solidsubstance, wherein said peak list includes the following data obtainedfrom the powder diffraction data: i. 2θ[deg]; ii. sin(θ); iii. κ[1/Å];iv. κ²[1/Å²] and/or v. d/n [Å], where κ=2 sin(θ)/λ and d/n=1/κ.
 4. Amethod as in claim 3, wherein the Triads Algorithm, comprising thefollowing steps, is performed: b. choosing three basis peaks from thepeak list and labeling them A, B, and C; c. using (001), (010), and(100), respectively, as Miller indices, to calculate the reciprocal celllength parameters (a*,b*,c*) from κ_(C), κ_(B), and κ_(A); d.determining acceptable ranges for the vector sums of pairs of C, B, andA using any of the following ranges:κ_(B)−κ_(C)≦κ_(B+C)≦κ_(B)+κ_(C),  i.κ_(A)−κ_(C)≦κ_(A+C)≦κ_(A)+κ_(C), and  ii.κ_(A)−κ_(B)≦κ_(A+B)≦κ_(A)+κ_(B);  iii. e. choosing three peaks from theranges in (d) and labeling as B+C, A+C, and A+B; f. using (011), (101),and (110), respectively, as Miller indices, to calculate the cosines ofthe reciprocal cell angle parameters of (α*,β*,γ*) from κ_(B+C),κ_(A+C), and κ_(A+B); g. calculating peak positions; and h. determiningwhether the cell is consistent with the entire peak list.
 5. A method asin claim 4, wherein step (c) is performed after step (e) rather thanafter step (b).
 6. A method as in claim 4, comprising the further stepof (i) refining the unit cell parameters.
 7. A method as in claim 6,further comprising one or more steps chosen from evaluating the trialunit cell, calculating the real space unit cell, calculating the reducedbasis, calculating the conventional unit cell, and identifying theextinction symbol.
 8. A method as in claim 6, further comprising one ormore of the following steps: j. calculating the real space cellparameters; k calculating the reduced basis parameters; l. refining thereduced basis parameters; m. identifying the character corresponding tothe reduced basis; n. calculating the conventional cell parameters; o.refining the conventional cell parameters; p. applying the extinctionsymbol; q. evaluating the solution and recording if favorable; and rpost-processing the data.
 9. A method as in claim 8, whereinpost-processing the data comprises comparing the data to data from asecond crystal form to determine whether the two samples represent thesame or different crystal forms.
 10. A method as in claim 9, comprisingthe steps of: i. indexing powder diffraction data from two differentsamples; and ii. comparing extinction symbols and/or numerical valuesfor the unit cell parameters for each sample.
 11. A method fordetermining the unit cell parameters of a crystalline solid form of acompound, which method comprises using an X-ray powder diffractometer togenerate XRPD data for a solid crystalline substance; and a. generatinga peak list from the XRPD data of the crystalline solid substance; b.choosing three basis peaks and labeling them A, B, and C; c. using(001), (010), and (100), respectively, as Miller indices, to calculatethe reciprocal cell length parameters (a*,b*,c*) from κ_(C), κ_(B), andκ_(A); d. determining acceptable ranges for the vector sums of pairs ofC, B, and A using any of the following ranges:κ_(B)−κ_(C)≦κ_(B+C)≦κ_(B)+κ_(C),  i.κ_(A)−κ_(C)≦κ_(A+C)≦κ_(A)+κ_(C), and  ii.κ_(A)−κ_(B)≦κ_(A+B)≦κ_(A)+κ_(B);  iii. e. choosing three peaks from theranges in (d) and labeling as B+C, A+C, and A+B; f. using (011), (101),and (110), respectively, as Miller indices, to calculate the cosines ofthe reciprocal cell angle parameters of (α*,β*,γ*) from κ_(B+C),κ_(A+C), and κ_(A+B); g. calculating peak positions; h. determiningwhether the cell is consistent with the entire peak list; and i.refining the unit cell parameter.
 12. A method as in claim 11, furthercomprising one or more of the following steps: l. calculating the realspace cell parameters; k. calculating the reduced basis parameters; l.refining the reduced basis parameters; m. identifying the charactercorresponding to the reduced basis; n. calculating the conventional cellparameters; o. refining the conventional cell parameters; p. applyingthe extinction symbol; q. evaluating the solution and recording iffavorable; and r. post-processing the data.
 13. A method fordistinguishing between crystalline solid forms of different samples of asubstance, which comprises: for each sample, generating an X-ray powderdiffraction pattern of a solid crystalline substance; determining theunit cell parameters of each substance by performing a Triads Algorithmto identify one or more sets of values of unit cell parameters of thecrystalline solid forms, and comparing the one or more sets of values ofunit cell parameters of the crystalline solid forms.
 14. A method ofsorting, screening, or ranking crystalline solid forms, which comprises:for each sample, generating an X-ray powder diffraction pattern of thesolid crystalline form; determining the unit cell parameters of thesubstance by performing a Triads Algorithm to identify one or more setsof values of unit cell parameters of the crystalline solid form, andsorting, screening, or ranking the crystalline solid forms based on theunit cell parameters, character, and/or extinction symbol of eachcrystalline solid form.